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The Canonical Ring of a Stacky Curve John Voight David Zureick-Brown

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The Canonical Ring of a Stacky Curve John Voight David Zureick-Brown The canonical ring of a stacky curve John Voight David Zureick-Brown Author address: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA Email address: [email protected] Department of Mathematics and Computer Science, Emory Univer- sity, Atlanta, GA 30322 USA Email address: [email protected] Contents Chapter 1. Introduction 1 1.1. Motivation: Petri's theorem 1 1.2. Orbifold canonical rings 1 1.3. Rings of modular forms 2 1.4. Main result 3 1.5. Extensions and discussion 4 1.6. Previous work on canonical rings of fractional divisors 6 1.7. Computational applications 6 1.8. Generalizations 6 1.9. Organization and description of proof 7 1.10. Acknowledgements 8 Chapter 2. Canonical rings of curves 9 2.1. Setup 9 2.2. Terminology 11 2.3. Low genus 14 2.4. Basepoint-free pencil trick 15 2.5. Pointed gin: High genus and nonhyperelliptic 16 2.6. Gin and pointed gin: Rational normal curve 19 2.7. Pointed gin: Hyperelliptic 20 2.8. Gin: Nonhyperelliptic and hyperelliptic 23 2.9. Summary 26 Chapter 3. A generalized Max Noether's theorem for curves 27 3.1. Max Noether's theorem in genus at most 1 27 3.2. Generalized Max Noether's theorem (GMNT) 29 3.3. Failure of surjectivity 30 3.4. GMNT: nonhyperelliptic curves 31 3.5. GMNT: hyperelliptic curves 32 Chapter 4. Canonical rings of classical log curves 35 4.1. Main result: classical log curves 35 4.2. Log curves: Genus 0 36 4.3. Log curves: Genus 1 36 4.4. Log degree 1: hyperelliptic 37 4.5. Log degree 1: nonhyperelliptic 39 4.6. Exceptional log cases 41 4.7. Log degree 2 42 4.8. General log degree 44 4.9. Summary 46 v vi CONTENTS Chapter 5. Stacky curves 49 5.1. Stacky points 49 5.2. Definition of stacky curves 50 5.3. Coarse space 51 5.4. Divisors and line bundles on a stacky curve 54 5.5. Differentials on a stacky curve 56 5.6. Canonical ring of a (log) stacky curve 58 5.7. Examples of canonical rings of log stacky curves 61 Chapter 6. Rings of modular forms 65 6.1. Orbifolds and stacky Riemann existence 65 6.2. Modular forms 67 Chapter 7. Canonical rings of log stacky curves: genus zero 69 7.1. Toric presentation 69 7.2. Effective degrees 73 7.3. Simplification 76 Chapter 8. Inductive presentation of the canonical ring 81 8.1. The block term order 81 8.2. Block term order: examples 82 8.3. Inductive theorem: large degree canonical divisor 83 8.4. Main theorem 87 8.5. Inductive theorems: genus zero, 2-saturated 87 8.6. Inductive theorem: by order of stacky point 89 8.7. Poincar´egenerating polynomials 95 Chapter 9. Log stacky base cases in genus 0 97 9.1. Beginning with small signatures 97 9.2. Canonical rings for small signatures 98 9.3. Conclusion 109 Chapter 10. Spin canonical rings 113 10.1. Classical case 113 10.2. Modular forms 115 10.3. Genus zero 116 10.4. Higher genus 117 Chapter 11. Relative canonical algebras 121 11.1. Classical case 121 11.2. Relative stacky curves 123 11.3. Modular forms and application to Rustom's conjecture 125 Appendix. Tables of canonical rings 127 Bibliography 139 Abstract Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an ex- plicit Gr¨obnerbasis. We work in a general algebro-geometric context and treat log canonical and spin canonical rings as well. As an application, we give an explicit presentation for graded rings of modular forms arising from finite-area quotients of the upper half-plane by Fuchsian groups. Received by the editor June 5, 2020. 2010 Mathematics Subject Classification. Primary 14Q05; Secondary 11F11 . Key words and phrases. Canonical rings, canonical embeddings, stacks, algebraic curves, modular forms, automorphic forms, generic initial ideals, Gr¨obnerbases. vii CHAPTER 1 Introduction 1.1. Motivation: Petri's theorem The quotient X = ΓnH of the upper half-plane H by a torsion-free cocompact Fuchsian group Γ ≤ PSL2(R) naturally possesses the structure of a compact Rie- mann surface of genus g ≥ 2; and conversely, every compact Riemann surface of genus g ≥ 2 arises in this way. The Riemann surface X can be given the struc- ture of a nonsingular projective (algebraic) curve over C: indeed, when X is not hyperelliptic, the canonical map X,! Pg−1 obtained from global sections of the sheaf Ω = ΩX of holomorphic differential 1-forms on X gives such an algebraic structure. Even when X is hyperelliptic, the canonical ring (sometimes also called the homogeneous coordinate ring) 1 M R = R(X) = H0(X; Ω⊗d) d=0 has X ' Proj R (as Ω is ample). Much more is known about the canonical ring: for a general curve of genus g ≥ 4, its image is cut out by quadrics. More specifically, by a theorem of Enriques, completed by Babbage [Bab39], and going by the name Petri's theorem [Pet23], if X is neither hyperelliptic, trigonal (possessing a map X ! P1 of degree 3), nor a plane curve of degree 5 (and genus 6), then R ' C[x1; : : : ; xg]=I is generated in degree 1 and the canonical ideal I of relations in these generators is generated in degree 2. In fact, Petri gives explicit quadratic relations that define the 0 ideal I in terms of a certain choice of basis x1; : : : ; xg for H (X; Ω) and moreover describes the syzygies between these quadrics. This beautiful series of results has been generalized in several directions. Arba- rello{Sernesi [AS78] considered embeddings of curves obtained when the canonical sheaf is replaced by a special divisor without basepoints. Noot [Noo88] and Do- dane [Dod09] considered several generalizations to stable curves. Another rich generalization is the conjecture of Green [Gre82], where generators and relations for the canonical ring of a variety of general type are considered. Green [Gre84] also conjectured a relationship between the Clifford index of a curve and the degrees of the subsequent syzygies (as a graded module) for the canonical ring; for curves of Clifford index 1 (trigonal curves and smooth plane quintics), this amounts to Petri's theorem. This second conjecture of Green was proved for a generic curve by Voisin; see the survey by Beauville [Bea05]. 1.2. Orbifold canonical rings Returning to the opening paragraph, though, it is a rather special hypothesis on the Fuchsian group Γ (finitely generated, of the first kind) that it be cocompact and torsion free. Already for Γ = PSL2(Z), this hypothesis is too restrictive, as 1 2 1. INTRODUCTION PSL2(Z) is neither cocompact nor torsion free. One can work with noncocompact groups by completing ΓnH and adding points called cusps, and then working with quotients of the (appropriately) completed upper half-plane H∗. We denote by H(∗) either the upper half-plane or its completion, according as Γ is cocompact or not, and let ∆ denote the divisor of cusps for Γ, an effective divisor given by the sum over the cusps. In general, any quotient X = ΓnH(∗) with finite area can be given the structure of a Riemann surface, but only after \polishing" the points with nontrivial stabilizer by adjusting the atlas in their neighborhoods. The object X itself, on the other hand, naturally has the structure of a 1-dimensional complex orbifold (\orbit space of a manifold"): a Hausdorff topological space locally modeled on the quotient of C by a finite group, necessarily cyclic. Orbifolds show up naturally in many places in mathematics [Sat56, Thu97]. So the question arises: given a compact, connected complex 1-orbifold X over C, what is an explicit description of the canonical ring of X? Or, put another way, what is the generalization of Petri's theorem (and its extensions) to the case of complex orbifold curves? This is the central question of this monograph. 1.3. Rings of modular forms This question also arises in another language, as the graded pieces 0 ⊗d Rd = H (X; Ω ) of the canonical ring go by another name: they are naturally identified with certain spaces of modular forms of weight k = 2d on the group Γ (see section 6.2). More generally, 0 ⊗d H (X; Ω(∆) ) ' M2d(Γ) is the space of modular forms of weight k = 2d, and so we are led to consider the canonical ring of the log curve (X; ∆), 1 M R(X; ∆) = H0(X; Ω(∆)⊗d); d=0 where ∆ is the divisor of cusps. For example, the group Γ = PSL2(Z) with X(1) = ΓnH∗ and ∆ = 1 the cusp at infinity has the ring of modular forms R(X(1); ∆) = C[E4;E6]; a graded polynomial ring in the Eisenstein series E4;E6 of degrees 2 and 3 (weights 4 and 6), respectively. Consequently, the log curve (X(1); ∆) is described by its canonical ring, and X(1) ' Proj R(X(1); ∆) as Riemann surfaces or as curves over C, even though X(1) has genus 0 and thus has a trivial canonical ring. In this way, the log curve (X(1); ∆) behaves like a curve with an ample canonical divisor and must be understood in a different way than the classical point of view with which we began. The calculation of the dimension of a space of modular forms using the valence formula already suggests that there should be a nice answer to the question above that extends the classical one. We record the relevant data in the signature of the Fuchsian group Γ ≤ PSL2(R): if Γ has elliptic cycles (conjugacy classes of elements of finite order) with orders 2 ≤ e1 ≤ · · · ≤ er < 1 and δ parabolic cycles 1.4.
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